[Paper Review] A classification of subsystems of a root system

Toshio Oshima|ArXiv.org|Nov 29, 2006

Advanced Algebra and Geometry4 citations

TL;DR

This paper provides a comprehensive classification of subsystems of root systems by analyzing homomorphisms that preserve Cartan integers, using Weyl group actions and dual pairs to distinguish isomorphism classes. The key contribution is a unified framework—based on extended Dynkin diagrams—for classifying subsystems, especially in exceptional root systems, with explicit tables detailing orbit counts and outer automorphism actions.

ABSTRACT

We classify isomorphic classes of the homomorphisms of a root system $Ξ$ to a root system $Σ$ which do not change Cartan integers. We examine several types of isomorphic classes defined by the Weyl group of $Σ$, that of $Ξ$ and the automorphisms of $Σ$ or $Ξ$ etc. We also distinguish the subsystem generated by a subset of a fundamental system. We introduce the concept of the dual pair for root systems which helps to study the action of the outer automorphism of $Ξ$ on the homomorphisms.

Motivation & Objective

• To classify all subsystems of a root system Σ up to Weyl group equivalence, particularly for exceptional types.
• To determine when isomorphic subsystems are equivalent under the Weyl group of Σ.
• To analyze the role of outer automorphisms of subsystems in realizing Weyl group actions.
• To distinguish subsystems generated by subsets of fundamental systems and compute their orbit counts.
• To provide a complete reference table of subsystem types with orbit counts under Weyl, automorphism, and outer automorphism groups.

Proposed method

• Uses the set of homomorphisms Hom(Ξ, Σ) that preserve Cartan integers to define subsystems isomorphic to Ξ.
• Reduces classification to graphic combinatorics on extended Dynkin diagrams, generalizing known results on graph automorphisms.
• Introduces the concept of dual pairs of subsystems to study the action of Aut(Ξ) on Hom(Ξ, Σ).
• Defines closures and S-closed subsystems to stabilize classification under successive root removal from extended Dynkin diagrams.
• Computes orbit counts via WΣ\Hom(Ξ,Σ)/Aut(Ξ) and related quotients to enumerate distinct equivalence classes.
• Uses the notation Σ^L for long roots and A_m^S/L, D_m^S/L to distinguish root lengths and subsystem types.

Experimental results

Research questions

• RQ1 Which abstract root systems Ξ can appear as subsystems of a given root system Σ?
• RQ2 Given two isomorphic subsystems Ξ and Ξ′ of Σ, when is Ξ′ equivalent to Ξ under the Weyl group WΣ?
• RQ3 How many WΣ-orbits of subsystems isomorphic to Ξ exist within Σ?
• RQ4 Does an outer automorphism of Ξ arise from an element of WΣ, particularly when Ξ has isomorphic irreducible components?
• RQ5 When is a subsystem Ξ equivalent to the subsystem generated by a subset of a fundamental system of Σ, and how many such subsets exist?

Key findings

• For irreducible Σ, the number of WΣ-orbits of Hom(Ξ, Σ) is often 1, indicating that all isomorphic subsystems are WΣ-equivalent.
• In cases like 4A₁ in E₇ and E₈, the orbit count under WΣ\Hom(Ξ,Σ)/Out(Ξ) is 2, with distinct classes labeled [Ξ]′ and [Ξ]′′.
• The dual pair construction enables analysis of outer automorphism actions on Hom(Ξ, Σ), especially when Ξ has multiple isomorphic components.
• For Ξ = D₅ + A₂ in E₈, OutΣ(Ξ) ≅ N_AutΣ(Ξ′)(Ξ₂)/WΞ, showing how stabilizer subgroups control outer automorphism realization.
• The table in Section 10 provides complete data: orbit counts, closures, and references to explicit constructions for all non-empty Hom(Ξ, Σ) with irreducible Σ.
• The method successfully generalizes Dynkin’s classification of regular subalgebras by using extended Dynkin diagrams and root-length distinctions (S/L), correcting earlier tables in [4].